§10.60 Sums

Keywords:: spherical Bessel functions
Referenced by:: §10.47(i)
Permalink:: http://dlmf.nist.gov/10.60

§10.60(i) Addition Theorems
§10.60(ii) Duplication Formulas
§10.60(iii) Other Series
§10.60(iv) Compendia

§10.60(i) Addition Theorems

Notes:: For (10.60.1)–(10.60.3) use (10.23.8) with $\nu=\tfrac{1}{2}$ and $\mathop{\mathscr{C}\/}\nolimits=\mathop{Y\/}\nolimits,\mathop{J\/}\nolimits,\mathop{{H^{{(1)}}}\/}\nolimits$ ; subsequently apply (10.47.12) and (10.47.13) in the case of (10.60.3).
Keywords:: spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.60.i

Define $u$ , $v$ , $w$ , and $\alpha$ as in §10.23(ii). Then with $\mathop{P_{{n}}\/}\nolimits$ again denoting the Legendre polynomial of degree $n$ ,

10.60.1 $\frac{\mathop{\cos\/}\nolimits w}{w}=-\sum _{{n=0}}^{\infty}(2n+1)\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(v\right)\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(u\right)\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right),$ $|ve^{{\pm i\alpha}}|<|u|$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind and $n$ : integer
A&S Ref:: 10.1.46 (corrected)
Referenced by:: §10.60(i)
Permalink:: http://dlmf.nist.gov/10.60.E1
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10.60.2 $\frac{\mathop{\sin\/}\nolimits w}{w}=\sum _{{n=0}}^{\infty}(2n+1)\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(v\right)\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(u\right)\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right).$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind and $n$ : integer
A&S Ref:: 10.1.45
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E2
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10.60.3 $\frac{e^{{-w}}}{w}=\frac{2}{\pi}\sum _{{n=0}}^{\infty}(2n+1)\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(v\right)\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(u\right)\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right),$ $|ve^{{\pm i\alpha}}|<|u|$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function and $n$ : integer
A&S Ref:: 10.2.35 (with condition added)
Referenced by:: §10.60(i)
Permalink:: http://dlmf.nist.gov/10.60.E3
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§10.60(ii) Duplication Formulas

Notes:: For (10.60.4) set $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits=\mathop{Y_{{\nu}}\/}\nolimits$ , $u=v=z$ , $\nu=-n-\frac{1}{2}$ , and $\alpha=\pi$ in (10.23.8). Then refer to (10.47.3), (10.47.4), and also apply the following results obtained from Table 18.6.1: $\mathop{C^{{(-n-\frac{1}{2})}}_{{k}}\/}\nolimits\!\left(-1\right)$ equals $\ifrac{(2n+1)!}{(k!(2n+1-k)!)}$ when $k=0,1,\dots,2n+1$ , and equals 0 when $k=2n+2,2n+3,\dots$ . For (10.60.5) use the same procedure, but with $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits=\mathop{J_{{\nu}}\/}\nolimits$ . Lastly, (10.60.6) follows by combining (10.60.4) and (10.60.5) with §10.47(iv).
Keywords:: spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.60.ii

10.60.4 $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(2z\right)=-n!z^{{n+1}}\sum _{{k=0}}^{n}\frac{2n-2k+1}{k!(2n-k+1)!}\mathop{\mathsf{j}_{{n-k}}\/}\nolimits\!\left(z\right)\mathop{\mathsf{y}_{{n-k}}\/}\nolimits\!\left(z\right),$

Symbols:: $!$ : $n!$ : factorial, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 10.1.49 (corrected)
Referenced by:: §10.60(ii), Ch.10
Permalink:: http://dlmf.nist.gov/10.60.E4
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10.60.5 $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(2z\right)=n!z^{{n+1}}\sum _{{k=0}}^{n}\frac{n-k+\frac{1}{2}}{k!(2n-k+1)!}{\left({\mathop{\mathsf{j}_{{n-k}}\/}\nolimits^{{2}}}\!\left(z\right)-{\mathop{\mathsf{y}_{{n-k}}\/}\nolimits^{{2}}}\!\left(z\right)\right)},$

Symbols:: $!$ : $n!$ : factorial, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §10.60(ii)
Permalink:: http://dlmf.nist.gov/10.60.E5
Encodings:: TeX, pMML, png

10.60.6 $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(2z\right)=\frac{1}{\pi}n!z^{{n+1}}\sum _{{k=0}}^{n}(-1)^{k}\frac{2n-2k+1}{k!(2n-k+1)!}{\mathop{\mathsf{k}_{{n-k}}\/}\nolimits^{{2}}}\!\left(z\right).$

Symbols:: $!$ : $n!$ : factorial, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 10.2.38
Referenced by:: §10.60(ii), Ch.10
Permalink:: http://dlmf.nist.gov/10.60.E6
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§10.60(iii) Other Series

Notes:: For (10.60.7)–(10.60.9) see Watson (1944, pp.368–369). For (10.60.10) use Watson (1944, p. 370, Eq. (9)) with $\nu=\tfrac{1}{2}$ , $\phi=\alpha$ , $\phi^{{\prime}}=\tfrac{1}{2}\pi$ ; also Eq. (18.7.9). For (10.60.11) see Watson (1944, p. 152). For (10.60.12) and (10.60.13) substitute $u=v=z$ , with $\alpha=0$ and $\pi$ , into (10.60.2). For (10.60.14) see Vavreck and Thompson (1984).
Permalink:: http://dlmf.nist.gov/10.60.iii

10.60.7 $e^{{iz\mathop{\cos\/}\nolimits\alpha}}=\sum _{{n=0}}^{\infty}(2n+1)i^{n}\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.47
Referenced by:: §10.60(iii), Ch.10
Permalink:: http://dlmf.nist.gov/10.60.E7
Encodings:: TeX, pMML, png

10.60.8 $e^{{z\mathop{\cos\/}\nolimits\alpha}}=\sum _{{n=0}}^{\infty}(2n+1)\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.2.36
Permalink:: http://dlmf.nist.gov/10.60.E8
Encodings:: TeX, pMML, png

10.60.9 $e^{{-z\mathop{\cos\/}\nolimits\alpha}}=\sum _{{n=0}}^{\infty}(-1)^{n}(2n+1)\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right).$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.2.37
Referenced by:: §10.60(iii), Ch.10
Permalink:: http://dlmf.nist.gov/10.60.E9
Encodings:: TeX, pMML, png

10.60.10 $\mathop{J_{{0}}\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\alpha\right)=\sum _{{n=0}}^{\infty}(4n+1)\frac{(2n)!}{2^{{2n}}(n!)^{2}}\mathop{\mathsf{j}_{{2n}}\/}\nolimits\!\left(z\right)\mathop{P_{{2n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $!$ : $n!$ : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.48
Referenced by:: §10.60(iii), Ch.10
Permalink:: http://dlmf.nist.gov/10.60.E10
Encodings:: TeX, pMML, png

10.60.11 $\sum _{{n=0}}^{\infty}{\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{2}}}\!\left(z\right)=\frac{\mathop{\mathrm{Si}\/}\nolimits\!\left(2z\right)}{2z}.$

Symbols:: $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.52
Referenced by:: §10.60(iii), Ch.10
Permalink:: http://dlmf.nist.gov/10.60.E11
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For $\mathop{\mathrm{Si}\/}\nolimits$ see §6.2(ii).

10.60.12 $\sum _{{n=0}}^{\infty}(2n+1){\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{2}}}\!\left(z\right)=1,$

Symbols:: $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.50
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E12
Encodings:: TeX, pMML, png

10.60.13 $\sum _{{n=0}}^{\infty}(-1)^{n}(2n+1){\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{2}}}\!\left(z\right)=\frac{\mathop{\sin\/}\nolimits\!\left(2z\right)}{2z},$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.51
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E13
Encodings:: TeX, pMML, png

10.60.14 $\sum _{{n=0}}^{\infty}(2n+1)({\mathop{\mathsf{j}_{{n}}\/}\nolimits^{{\prime}}}\!\left(z\right))^{2}=\tfrac{1}{3}.$

Symbols:: $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E14
Encodings:: TeX, pMML, png

For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000).

§10.60(iv) Compendia

Keywords:: spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.60.iv

For collections of sums of series relevant to spherical Bessel functions or Bessel functions of half odd integer order see Erdélyi et al. (1953b, pp. 43–45 and 98–105), Gradshteyn and Ryzhik (2000, §§8.51, 8.53), Hansen (1975), Magnus et al. (1966, pp. 106–108 and 123–138), and Prudnikov et al. (1986b, pp. 635–637 and 651–700). See also Watson (1944, Chapters 11 and 16).

§10.60 Sums

Contents

§10.60(i) Addition Theorems

§10.60(ii) Duplication Formulas

§10.60(iii) Other Series

§10.60(iv) Compendia